Small pore theory

An interesting subcase of the above is the small pore theory, sometimes known as Schmid theory [61. Unlike the preceding large pore theory, in which the assumption Is made that the diffuse layer thickness is small compared to the capillary diameter, here the excess of ions is distributed throughout the entire void volume [24], i.e. it is assumed that the double layers overlap. Hence, the electrical force can act more uniformly across the pore section, as is the case for an external hydraulic pressure. This condition is treated also by FUce and Whitehead [16], for example. The equation for the velocity, when xa 1, becomes... 

Velocity profiles across the capillary have a Poisseuille shaped flow and the expression predicts that the electroosmotic coefficient of permeability should vary with the square of the radius. In practice, it is found generally that this law is not as satisfactory as the Helmholtz-Smoluchowski approach for predicting electroosmotic behavior in soils. The failure of small pore theory may be because most clays have an aggregate structure with the flow determined by the larger pores [6], Another theoretical approach is referred to as the Spiegler Friction theory [25,6]. Its assumption, that the medium for electroosmosis is a perfect permselective membrane, is obviously not valid for soils, where the pore fluid comprises dilute electrol d e. An expression is derived for the net electroosmotic flow, Q, in moles/Faraday,... 

With respect to the size and charge selectivity of paracellular pathways, equivalent pore theory has been utilized to calculate an effective radius based on the membrane transport of uncharged hydrophilic molecules, while equivalent circuit theory has been used to separate mediated from paracellular membrane transport of small ions. The term equivalent should be emphasized, as selectivity parameters are obtained from membrane transport data, so phenomenological information is used to quantitate the magnitude of aqueous pathways... 

Information relating to the diffusion of metal-bearing compounds in catalytic materials at reaction conditions has been obtained indirectly through classic diffusion and reaction theory. Shah and Paraskos (1975) calculated effective diffusitivities of 7 x 10-8 and 3 x 10-8 cm2/sec for V and Ni compounds in reduced Kuwait crude at 760°F. These low values may be indicative of a small-pore HDS catalyst. In contrast, Sato et al. (1971) report that the effective diffusivity of vanadium compounds was one-tenth that of the nickel compounds on the basis of metal deposition profiles in aged catalysts. This large difference may be influenced by relative adsorption strengths not explicitly considered in their analysis. 

In the following investigation, we use this theory allowing a continuum thermomechanical approach to the two-phase flow problem in a porous solid. The transient and stationary motion of liquids in porous solids with small pores is complex and not all related problems have been solved yet. The main internal... 

The applicability of the Saam-Cole theory has been tested by Findenegg and his co-workers (1993,1994). Their adsorption measurements of certain organic vapours on carefully selected grades of controlled-pore glass provide semi-quantitative confirmation of the theoretical treatment adopted so far. However, it is evident that some refinement is required in the assessment of 0(rp) for materials of small pore size and that the experimental choice of the mesoporous adsorbent is important To make further progress it will be necessary to study adsorbents having narrow size and shape distributions of easily accessible mesopores. 

Most of these theories have one significant drawback They are all derived for the flat open adsorbent surface, but HPLC adsorbents are porous materials with average pore size on the level of 100 A for bare material. After chemical modification of the original silica surface, the effective pore diameter decreases and the properties of electric double layer in the confined space of small pores are significantly different from that on the flat surface. 

The relationship between separation properties and casting parameters depends on the membrane structure. According to the Bokhorst - Altena — Smolders theory of phase separation [ 8 ], when PS concentration in the casting solution is increased and the time of solvent evaporation is extended, pore diameter decreases, thus improving the selectivity of the membranes. However, the increase of temperature to a critical value accounts for the increase of pore diameter, which brings about a decrease of the separation factor value. Further increase in temperature brings about a rapid evaporation of the solvent to yield small pore diameter membranes characterized by better separation properties. 

A Bethe-tree is a particular case of more general networks considered in percolation theory. Sahimi and Tsotsis [1985] applied percolation theory and Monte Carlo simulation to deactivation in zeolites, approximated by a simple cubic lattice. Beyne and Froment [1990, 1993] applied percolation theory to reaction, diffusion and deactivation in the real ZSM-5 lattice. The finite rate of growth was described in terms of a polymerization mechanism. Pore blockage was reached in this small pore zeolite. It also affects the path followed by the diffusing molecules that becomes more tortuous, so that the effective diffusivity has to be expressed in terms of the blockage probability. 

Of the established static techniques, which we have considered here, that involving gas adsorption isotherm measurements remains one of the most powerful and widely applicable. It is indeed very accessible with the availability of automated commercial equipment and the variety of data treatment facilities available. Nevertheless, it is still circumscribed by the assumptions implicit in the choice of a pore shape model in the case of mesoporous materials. Its application to microporous structures has recently advanced considerably, although there are here certain reservations which still exist concerning the general application of theories to describe adsorption in such small pores in ill defined structures. 

One limitation of the single-events theory consists in considering the surface intermediates, carbocations and/or activated compounds, as if they were free species or radicals. In the important case of small-pore zeolitic catalysts, this limited representation of the catalytic act is no longer valid since the shape selectivity problems, introduced by the geometry and confinement of zeolitic cages, are ignored. 

Desorption is often a slower process than adsorption, and a phenomenon called aging prevents a total extraction by water or mild extraction solutions (methanohwater (1 1 v/v), ammonium acetate, etc.) to extract the pesticide completely from soil, even if it is sufficiently soluble in the solvent. The amount that is nonextractable increases over time. The reason for this has not yet been settled. One theory is that the soil has many extremely small pores (nanopores). In the course of time the adsorbed molecules diffuse into such pores, where they are not easily extracted and have a low and decreasing bioavailability. An alternative theory is that they are first adsorbed to some easily available low-affinity binding sites and over time jump over to high-affinity binding sites. 

In each of the above analyses, the pores were considered as parallel sets of large and small pores without interconnection between the separate sets. However, most void structures comprise a network in interconnected void spaces and "network effects" will diaate the potential implications of changes in pore structure. The generic influence of pore networks were analyzed by Beeckman and Froment22 based on modified Bethe tree two-dimensional networks. Based on this simulated analyses, the authors concluded that the nature of the deactivation does depend on the nature of the network structure. Sahami and Tsotsis employed percolation theory to analyze a three-dimensional network of interconnected pores and concluded that the void interconnectivity is crucial in determining the influence of network structure on the deactivation phenomena. 

This technique can be considered a standard method in the science of porous ceramics and catalysts. It is based on the principle that inside a small pore a gas can condense to a liquid at a relative pressure lower than unity this introduces the capillary condensation theory. The adsorption and desorption isotherms of an inert gas are determined as a function of the relative pressure (prei = pIpQ, i.e., the ratio between the applied pressure and the saturation pressure). N2 is often used as adsorption gas, and the experiments are carried out at the boiling liquid nitrogen temperature (at 1 bar). The adsorption isotherm... 

Beside classical methods of pore size analysis, there are many advanced methods. Seaton et al. [161] proposed a method based on the mean field theory. Initially this method was less accurate in the range of small pore sizes, but even so it g ve a more realistic -way for evaluation of the pore size distribution than the classical methods based on the Kelvin equation [162]. More rigorous methods based on molecular approaches such as grand canonical Monte Carlo (GCMC) simulations [147, 163-165] and nonlocal density functional theory (NLDFT) [86, 146, 147, 161, 163-169] have been developed and their use for pore size analysis of active carbons is continuously growing. 

At this time it had become possible to determine experimentally total surface area and the distribution of sizes and total volume of pores. Wheeler set forth to provide the theoretical development of calculating the role of this pore structure in determining catalyst performance. In a very slow reaction, reactants can diffuse to the center of the catalyst pellet before they react. On the other hand, in the case of a very active catalyst containing small pores, a reactant molecule will react (due to collision with pore walls) before it can diffuse very deeply into the pore structure. Such a fast reaction for which diffusion is slower than reaction will use only the outer pore mouths of a catalyst pellet. An important result of the theory is that when diffusion is slower than reaction, all the important kinetic quantities such as activity, selectivity, temperature coefficient and kinetic reaction order become dependent on the pore size and pellet size with which a pellet is prepared. This is because pore size and pellet size determine the degree to which diffusion affects reaction rates. Wheeler saw that unlike many aspects of heterogeneous catalysis, the effects of pore structure on catalyst behavior can be put on quite a rigorous basis, making predictions from theory relatively accurate and reliable. 

---